I totally agree with Steve. Connecting LEDs in parallel is SOMETIMES workable but not likely in this situation. Small differences between the LEDs will mean that some LEDs draw more current than others; they will get hotter, and this will reduce their forward voltage, so they will get hotter, and so on. This is called thermal runaway. Eventually you will have a situation where one LED (or perhaps a few LEDs) draw almost all of the current, and this LED will fail. Hopefully it will fail short, and protect the other LEDs.
Each LED should have its own current limiting resistor. This is my very strong recommendation. If you think it's not important, then think again. You are a beginner here; you've said so. We are not making up rules just to make your job harder. This is a serious issue.
You want to run each LED at 350 mA. These LEDs have a typical forward voltage of 3.5V and you're running them from a 5V supply. That leaves about 1.5V across each series resistor. So the calculation for the series resistor is R = V / I which is 1.5 / 0.35 which is 4.3 ohms. Power dissipation in each resistor will be 525 mW so you should use resistors rated for 1W and make sure they have some form of heatsinking.
As for driving them, the obvious choice would be an N-channel MOSFET with a logic-level gate, with a low ON-resistance. Have a look at
http://www.digikey.com/product-detail/en/NTD4906N-35G/NTD4906N-35GOS-ND/2194521. It may seem like overkill, being rated for 10A maximum current, but it has a very low Rds(on) (8 milliohms at Vgs=4.5V) and is among the cheapest suitable parts I could find - cheaper than many other parts with poorer specifications.
This device will switch your LED array ON and OFF under control of its gate, which is driven with a 0V/5V signal from the timer/dimmer circuitry. This signal can be pulse-width modulated (see
http://en.wikipedia.org/wiki/Pulse-width_modulation) to vary the LED brightness. You will need a good power supply to be able to supply all those LEDs. Also, avoid long wires, as PWM at high current will cause radiated EMI (interference).
Edit: This MOSFET is excellent for rapid switching of high-current loads with very low ON-resistance but it MUST be driven by a high-current driver so the transition through its linear region is as quick as possible. It is only rated to dissipate 2.6 watts so it must be switched extremely quickly between fully OFF and fully ON, and vice versa.